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It describes these ideas, and the rules that apply to them, by using letters to represent numbers. Algebra is used by software engineers who design computer programs and video games; civil engineers who design roads, railways and bridges; tradespeople who work on building sites; and many others.
28
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Page 1: Chapter 2

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2 PATTERNS AND ALGEBRA

Algebra is a shorthand way of writing many mathematical and scientifi c ideas. It describes these ideas, and the rules that apply to them, by using letters to represent numbers. Algebra is used by software engineers who design computer programs and video games; civil engineers who design roads, railways and bridges; tradespeople who work on building sites; and many others.

02 CO NCM8_4 SB TXT.indd Ch02:3402 CO NCM8_4 SB TXT.indd Ch02:34 19/9/08 12:40:43 PM19/9/08 12:40:43 PM

Page 2: Chapter 2

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In this chapter you will:

• translate between words and algebraic symbols, and between algebraic symbols and words

• substitute into algebraic expressions• generate a number pattern from an algebraic

expression• add and subtract like terms to simplify algebraic

expressions• simplify algebraic expressions that involve

multiplication and division• simplify algebraic expressions that involve adding

and subtracting algebraic fractions• (Stage 5) simplify algebraic expressions that

involve multiplying and dividing algebraic fractions• expand algebraic expressions by removing

grouping symbols• factorise algebraic terms and expressions• (Stage 5) apply the index laws to simplify

algebraic expressions

Wordbank

• algebraic expressions expressions that describe a quantity, using variables and numerals, such as 3x + 4y − 8

• algebraic terms the parts of an algebraic expression, such as 3x

• evaluate to fi nd the value of an algebraic expression after substitution

• expand to rewrite an expression such as 4(2k + 4) without grouping symbols

• factorise the opposite of expand; to rewrite an expression, such as 8k + 16, with grouping symbols

• like terms terms with exactly the same variables, such as 5p and 2p

• substitution replacing a variable with a number• variables letters of the alphabet used to

stand for a number, also called pronumerals or unknowns

02 CO NCM8_4 SB TXT.indd Ch02:3502 CO NCM8_4 SB TXT.indd Ch02:35 19/9/08 12:41:13 PM19/9/08 12:41:13 PM

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36 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

Start upWorksheet2-01

Brainstarters 2

1 Write each of these in the simplest way:a 7 × p b a × bc k × k d n × n × ne 2 × p × m f p × k × dg y × 6 h t × 3 × qi 1g j q + q + qk a + a + a + a l 1x

2 If x = 8 and y = 2, evaluate:a x + y b 3xc x − y d 5ye xy f 9 − yg x + 17 h 2x + 4i 3y − 2 j x ÷ yk 5x ÷ y l 4x + 5y

3 Find:a 3 − 4 b -5 + 9 c -8 − 2d -3 × 6 e -12 ÷ (-4) f -12 ÷ 4 − 10

4 Find:

a b c

d e f

2-01 From words to algebraic expressionsIn algebra, a variable (or pronumeral) is a letter of the alphabet used to stand for a number. It is called a variable because its value can vary (change). In this section, you will improve your ability to translate between words and algebraic symbols, and between algebraic symbols and words.

Skillsheet2-01

Algebraic expressions

38--- 1

8---+ 1

2--- 1

4---– 3

10------ 2

5---+

23--- 1

5---× 3

4--- 4

9---× 3

4--- 3

8---÷

Worksheet2-02

What’s theexpression?

Example 1

If N is a variable representing any number, write an algebraic expression for:a six times the number b the number divided by three

Solution

a 6 × N = 6N b N ÷ 3 or N3----

02 NCM8_4 SB TXT.fm Page 36 Friday, September 19, 2008 10:54 AM

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37ISBN: 9780170136952 CHAPTER 2 ALGEBRA

1 Let N represent any number. Write expressions for:a four times the number b the number plus threec the number divided by M d the number that comes after Ne half the number f seven times the numberg the sum of the number and 15 h twelve minus the numberi the number that comes before N j double the numberk the number divided by 4 l twice the number plus sevenm the product of the number and 10 n seven times the number minus twelve

2 If p, q and r represent any three numbers, write expressions for:a the sum of p, q and rb the product of p and qc the difference between r and q, where r is greater than qd the product of p and r, minus qe three times q, plus pf q divided by rg seven more than (p times r)h p times q divided by ri four times p, plus q multiplied by rj the product of p, q and r, decreased by seven times p

3 Write an algebraic expression for the answer to each of the following.a Think of a number (L), multiply it by 5 and add 2.b Think of a number (t), multiply it by 3 and then subtract 9.c Think of a number (w), multiply it by 2 and then add 17.d Think of a number (k), multiply it by 1.5 and take away 4.2.e Think of a number (x), then divide it by 5 and add 3.f Think of a number (A), halve it and then take away 12.g Think of a number (R ), divide it by 3 and then add 6.5.h Think of a number (y), multiply it by 5 and then divide it by 3.

Exercise 2-01

Example 2

Think of a number, L, multiply it by two and add seven.Write an algebraic expression for the final answer.

SolutionL × 2 + 7 = 2L + 7

Ex 1

Ex 2

02 NCM8_4 SB TXT.fm Page 37 Friday, September 19, 2008 10:54 AM

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38 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

4 Describe each of the following algebraic expressions in words:a x + y + z b 3N + Z c xy − 5 d

e 4x − 2 f xyz g 5N + 12 h 47 − 2N

2-02 Substitution

1 If m = 4 and n = 7, which of the following is the value of 2m − n? Select A, B, C or D.A 17 B 1 C -6 D 15

2 If a = 4, b = 7 and c = 2, evaluate each of the following:a a + b b 3a c a2 d a + b + ce bc f 4b − 2 g c + a − b h 7a − 2bi 20 − b j ab + c2 k b − c l 3a ÷ c

3 If j = 10, k = -3 and l = 5, evaluate each of the following:a jkl b j2 c k2 d j ÷ le jk ÷ 6 f k2 + l2 g 5k h 3j + 2k − l

i j + l − k j k 6jk l

4 Complete these tables using the given rules.a m = 6e b j = c + 4

c g = 3w + 10 d y = + 8

Exercise 2-02

N4----

Worksheet2-03

Tables of values 1

Worksheet2-04

Formula 1 game

Example 3

If m = 5 and n = 12, evaluate:a 2m + n b mn ÷ 4

Solutiona 2m + n = 2 × 5 + 12 b mn ÷ 4 = 5 × 12 ÷ 4

= 22 = 15

Ex 3

j2--- 2k–

j l+3

---------

e 0 5 9 23

m

c 4 8 11 32

j

x7---

w 1 3 11 32

g

x 7 14 28 84

y

02 NCM8_4 SB TXT.fm Page 38 Friday, September 19, 2008 10:54 AM

Page 6: Chapter 2

39ISBN: 9780170136952 CHAPTER 2 ALGEBRA

e y = 2x − 4 f n = − 1

g y = 4x − 3 h n = 2m + 1

m2----

x 5 10 12 20

y

m 4 16 21 25

n

x -4 -1 0 3

y

m 0 1

y

12--- 3

4--- 1

2---

Using technology

Substitution using spreadsheets1 Enter the values shown on the right into

a spreadsheet.

2 Use the given values to create spreadsheet formulas for the algebraic expressions following.a In cell D1, enter the formula for p − u (as shown below).

b In cell D2, enter the formula for m + n (as shown on the right).

c In cell D3, enter the formula for p × q.d In cell D4, enter the formula for t ÷ u × s.e In cell D5, enter the formula for n + t − m.f In cell D6, enter the formula for n − m ÷ u.

g In cell D7, enter the formula for

h In cell D8, enter the formula for s ÷ p × m.i In cell D9, enter the formula for p2.j In cell D10, enter the formula for (s + n) × (p + r).k In cell D11, enter the formula for t + p × q + s.l In cell D12, enter the formula for (s ÷ u)3.m In cell D13, enter the formula for m × n × p × q × r × s × t × u.n In cell D14, enter the formula for the average of m, n, p and q.

o In cell D15, enter the formula for v

This meansm = 10.

n m–u

------------- .

n p t+ +m q–

-------------------- .

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40 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

2-03 Adding and subtracting algebraic termsOnly like terms can be added or subtracted. Like terms are terms that have exactly the same variables (pronumerals). This means 2m and 5m are like terms. So are 4ab and 2ba. But 2d and 5m are not like terms. Nor are 3x and 5x2.

1 Select the like terms each time:a 4x, 2y, 2x, 5z, x b m, m2, n, 3m2, mn c 2ab, 4a, 6ab, 3b, 5d vw, 2v, 4v, w, v2, v e p, 6pq, 2qp, 3pq, 7q f 2c, 5, 3c2, 7, 9, 3d, 2

2 Simplify:a 6m − 2m b 3ab + 7ab c p + p + pd 12mn − 4mn e 7xy − 5xy f 8k − 3k − 4kg 4cd + 10cd − 3cd h m + 4m − m i 12y − 2y − 4yj 2x2 + 3x2 + x2 k 8p − 5p + 12p l 17rs − 12rs − 2rs

Exercise 2-03

p In cell D16, enter the formula for q In cell D17, enter the formula for s2 − 3.r In cell D18, enter the formula for 2n2 + r3.

3 Think of some different values for column B and use column E to create at least 10 different algebraic expressions using the symbols practised so far. Check if your answers are as you expected or if you need to insert grouping symbols to make them mathematically correct.

n p+ .

Worksheet2-05

Collecting like terms

Example 4

Simplify:a 8m + 2n + 3m + n b 6m − 8p + 3m + 6p

c 6xy + 7x2 − 3yx − 12x2

Solutiona 8m + 2n + 3m + n

= 8m + 3m + 2n + n Finding like terms.

= 11m + 3nb 6m − 8p + 3m + 6p Remember: The sign goes with the term that follows it.

= 6m + 3m − 8p + 6p= 9m − 2p

c 6xy + 7x2 − 3yx − 12x2

= 6xy − 3yx + 7x2 − 12x2

= 3xy − 5x2

02 NCM8_4 SB TXT.fm Page 40 Friday, September 19, 2008 10:54 AM

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41ISBN: 9780170136952 CHAPTER 2 ALGEBRA

3 Which of the following is 2k + 4j − 5k − 2j simplified? Select A, B, C or D.A -kj B 7k + 6 j C -3k + 2j D 13kj

4 Simplify:a 5k − 2j + 6j b 10ab + 2 − abc 12m2 − 5m2 + 2m d 12s2 − 7st − 2s2

e 4mn + 7mn − mn f 14k + 3l − 5kg y2 + 2y2 − y2 h 9d − 4e + 3di 7 − 2x − 3 j 15gh − 8gh + ghk 4p − 2q + 5q l 5a + 6c − 4c

5 Simplify each of the following:a 4x − y − x − 2y b 7ab + 2mn − 3ab + mnc 4k2 + 3 + 5k2+ 8 d 4bc − 8a + 2bc − ae x2 − 6x − x + 7 f p2 + q2 + 3p2 − q2

g 3 + 4y + 8 + 7y h 11r − 3s − 4r + 6si 7g + 8h − h − g j 6x2 − 4x − 9x + 6k 20a + 11m + 32a + 19m l 3q − 7 − 8q + 12m 7x + 2y + 8x − 4y n r2 + 4s2 + 12r2 − s2

2-04 Multiplying and dividing algebraic termsWhen multiplying or dividing algebraic terms, we multiply or divide the numbers first and then the variables.

Ex 4

Worksheet2-06

Perimeter and area

Skillsheet2-02

Algebra using diagrams

Worksheet2-07

Simplifying algebraic expressionsExample 5

Simplify:a 7m × 3m × 2 b 45m ÷ (-9)

Solutiona 7m × 3m × 2 = 7 × 3 × 2 × m × m

= 42m2

b 45m ÷ (-9) =

= -5m

45m-9

-----------

Example 6

Simplify by expressing in index form:a q × q × q b j × j × j × j × j × j

Solutiona q × q × q = q3 b j × j × j × j × j × j = j6

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42 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 Which of the following is 3k × 7kj simplified? Select A, B, C or D.A 21kj B 10k2j C 10kj D 21k2j

2 Simplify each of the following:a 4 × 3y b 8 × 7mc 2k × q d 4q × 7e 3 × f × 5 f 3p × 7qg 12p × 2m h 5a × 6ci 2y × 4y j 10 × 3m × 6nk 2x × 3x × 9 l 2r × 2 × 2r

3 Which of the following is simplified? Select A, B, C or D.

A 11mn B 11m2n C 50mn D 11m

4 Simplify each of the following:a 24m ÷ 6 b 36d ÷ 9

c 55k ÷ 11 d

e f

g 25f ÷ 5f h 10d ÷ d

i 14q ÷ 2q j

k l

5 Simplify each of the following:a -4b × 2d b 6m × (-4m)c -2 × 5k × 9l d -16a ÷ 4e 49k ÷ (-7) f -35b ÷ (-5b)g 3m × mn × 4n h 6abc ÷ 2abi 5r × (-2r) × (-4) j 9d × 4e ÷ 6dk 8x × 9x ÷ 12 l 108m2 ÷ 12m × 2n

6 Simplify:a y × y × y × y b a × a × ac 2 × c × c d t × t × t × t × te q × q × q × 7 f k × k × k × l × lg m × m × m × n × n h p × p × p × p × k × k × ki 3 × y × y × y j d × d × e × e × fk q × q × r × r × r × r × 4 l m × n × m × p

Exercise 2-04

Ex 5

55m2n5m

-----------------

72a9

---------

90s6

-------- 12kk

---------

63n7n

---------

20mn4

-------------- 32ef8

----------

Ex 6

02 NCM8_4 SB TXT.fm Page 42 Friday, September 19, 2008 10:54 AM

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43ISBN: 9780170136952 CHAPTER 2 ALGEBRA

2-05 Adding and subtracting algebraic fractions

Algebraic fractions follow the same rules as numerical fractions.To add or subtract algebraic fractions, convert the denominators so that they are the same.

Working mathematically

Equivalent expressions1 Determine whether each statement below is true or false by substituting a number

(of your own choice) for the variable and checking whether the left-hand side equals the right-hand side.a 6 + 4m + 4 = 10 + 4mb 5x + 3x = 8x2

c 4k − 7k = 3kd 7g + 8h − h − g = 7h + 6ge 3 + 7n + 13 + 2n = 25nf 3p × 7q = 21pqg 21m ÷ 3m = 7mh 2y × 4y = 24y

i = -4a

j 2r × 2 × 2r = 8r2

2 Explain why each of the false statements in Question 1 is incorrect (say what mistake has been made), and provide a correct statement.

-16a4

------------

Applying strategies and reasoning

Example 7

Simplify:

a b

Solution

a =

=

b =

=

=

m5---- 2m

5--------+ 7k

10------ 3k

10------–

m5---- 2m

5--------+ m 2m+

5-------------------

3m5

--------

7k10------ 3k

10------– 7k 3k–

10------------------

4k10------

2k5

------

02 NCM8_4 SB TXT.fm Page 43 Friday, September 19, 2008 10:54 AM

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44 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 Simplify:

a b c d

e f g h

i j k l

2 Simplify:

a b c d

e f g h

i j k l

Exercise 2-05

Example 8

Simplify:

a b

Solutiona Method 1:

Common denominator = 4 × 8 = 32

= = = =

=

=

=

Method 2:LCM of 4 and 8 = 8

= =

= +

=

b Method 1:Common denominator = 6 × 10 = 60

= = = =

=

=

=

Method 2:LCM of 6 and 10 = 30

= = = =

=

=

=

f4--- f

8---+ 5t

6----- 3t

10------–

f4--- f 8×

4 8×------------ 8f

32------ , f

8--- f 4×

8 4×------------ 4f

32------

f4--- f

8---+ 8f

32------ 4f

32------+

12f32--------

3f8-----

f4--- f 2×

4 2×------------ 2f

8-----

f4--- f

8---+ 2f

8----- f

8---

3f8-----

5t6----- 5t 10×

6 10×----------------- 50t

60-------- , 3t

10------ 3t 6×

10 6×--------------- 18t

60--------

5t6----- 3t

10------– 50t

60-------- 18t

60--------–

32t60--------

8t15------

5t6----- 5t 5×

6 5×-------------- 25t

30-------- , 3t

10------ 3t 3×

10 3×--------------- 9t

30------

5t6----- 3t

10------– 25t

30-------- 9t

30------–

16t30--------

8t15------

Ex 7

m2---- m

2----+ 7n

10------ 3n

10------– 2k

3------ k

3---+ 2x

3------ x

3---–

3d5

------ d5---– 5f

8----- 7f

8-----+ 3c

10------ 4c

10------+ 5x

8------ x

8---–

7m20-------- 11m

20-----------+ r

4--- 3r

4-----+ 7a

2------ 3a

2------– 4k

3------ 8k

3------+

Ex 8

f2--- f

3---+ 2q

3------ q

5---+ 3l

4----- l

2---– 5e

9----- e

2---–

7k8

------ k3---+ 4m

5-------- 3m

4--------– 9h

10------ 3h

5------– 5n

6------ 3n

8------+

k12------ 2k

3------+ 7d

8------ 3d

4------– 2t

3----- t

6---– 11m

10----------- 5m

4--------+

02 NCM8_4 SB TXT.fm Page 44 Friday, September 19, 2008 10:54 AM

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45ISBN: 9780170136952 CHAPTER 2 ALGEBRA

Stage 52-06 Multiplying and dividing algebraic fractions

You will recall from working with numerical fractions in Chapter 1:• when multiplying fractions, we multiply the numerators and multiply the denominators• when dividing by a fraction, we multiply by its reciprocal• in both cases, we cancel any common factors before calculating the answer.These same rules apply to algebraic fractions.

1 Simplify each of the following:

a b c d

e f g h

i j k l

m n o

Exercise 2-06

Example 9

Simplify

Solution

=

=

= 2

Cancelling common factors in the numerator and denominator.

6k--- 5k

12------× .

6k--- 5k

12------× 61

k1----- 5k1

122---------×

52---

12---

Example 10

Simplify

Solution

=

=

Multiplying by the reciprocal of

pq36------ 7p

12------÷ .

pq36------ 7p

12------÷ pq1

363-------- 121

7p1--------×

q21------

7p12-------.

Ex 9

a4--- c

6---× m

7---- n

4---× 3

x--- 9

y---× 7

p--- q

4---×

2a5

------ 311k---------× 5

x--- y

10------× a

9--- 4

a---× p

q-- q

r--×

t7--- 7

t---× 4

m---- 3m

6--------× 5k

4------ 8

15k---------× km

4n------- 2n

m------×

36d7

--------- 21d12

---------× 25x------ 10x

3---------× 77j

20k--------- 35k

11j---------×

02 NCM8_4 SB TXT.fm Page 45 Friday, September 19, 2008 10:54 AM

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46 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

2 Simplify each of the following:

a b c

d e f

g h i

j k l

m n o

2-07 Mixed problems

1 Simplify:a x + y + x + 2y b 4p + 7 + p + 3c 4a + 3b − 2a − b d 8m + 4n − 3m + 6ne x2 + y + 4x2 + 3y f 7m2 + 3n − 2m2 + 6ng 5fg − 2mk − 5fg + 4mk h 8ab − 6b + 2ab − 2bi 4m2 + 3m + 2m2 − m j 4x − y − x − 2y − yk 5a2 − 3a + 4a2 − 3a l -12 + 2q − 8 + 5q + 4

2 Simplify:a 4a × 2 b 16x ÷ 8 c 9 × 3md 22k ÷ 2 e 7 × 3p f 2c × cg 4k ÷ k h 12y × 2y i 7n ÷ nj 15f ÷ 3f k 3r × 5s l 4fg ÷ fm 3a × 5b × c n 16pqr ÷ 16q o 9m × 3n × 4

p q r

Exercise 2-07

Ex 10 m2---- m

7----÷ a

4--- 7

12------÷ 3

4--- 9

x---÷

15k

------ 52---÷ m

n---- p

q--÷ 6

n--- 5

n---÷

32x------ 1

3x------÷ cd

5----- d

10------÷ t

5--- t

5---÷

mnp

-------- nmp-------÷ d

2--- ad

6-----÷ 3k

7j------ k

14j--------÷

10x3y

--------- 5x6y------÷ 4ab

7--------- 2a

21------÷ 6 3

2a------÷

Remember:• only like terms can be added or subtracted• when multiplying or dividing, do the numbers first, then the variables• when substituting, replace the variable (letter) with the number and evaluate the answer• when adding or subtracting algebraic fractions, convert them to fractions with the same

denominator first.

!

14mn2m

-------------- 12a 2bc×8ac

----------------------- 8a 3ak×6k 2a×

----------------------

Stage 5

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47ISBN: 9780170136952 CHAPTER 2 ALGEBRA

3 If m = 9, n = -3 and p = 6, find:a m + n + p b mn − p c 6m + 2pd m ÷ n e n2 + p2 f mnpg 5p ÷ n h 10m ÷ p i m − n − pj mp − 30 k (m + n) ÷ p l (p − n − m) ÷ 7

4 Simplify:

a b c

d e f

g h i

2-08 Expanding algebraic expressionsAlgebraic expressions involving grouping symbols (brackets) can be rewritten without grouping symbols. This is called expanding the expression.The quick way to do this is to multiply every term inside the grouping symbols by the term outside the grouping symbols.

m4---- 2m

4--------+ 7k

10------ 2k

10------– 5a

8------ 3a

8------+

y3--- y

4---+ w

3---- w

7----– 5k

6------ 5k

12------–

l2--- 3l

4-----+ x

6--- 3x

8------+ 3n

4------ n

6---–

Worksheet2-08

Expandominoes

Skillsheet2-02

Algebra usingdiagrams

Example 11

Expand:a 7(p + 4) b -3(l − 2)

Solution

a 7(p + 4) = 7 × p + 7 × 4= 7p + 28

b -3(l − 2) = -3 × l − 3 × (-2)= -3l + 6

Example 12

Expand and simplify:a 4(x + 7) + 2(x − 4) b 5p − (p + 2)

Solutiona 4(x + 7) + 2 (x − 4) = 4 × x + 4 × 7 + 2 × x + 2 × (-4)

= 4x + 28 + 2x − 8= 6x + 20

b 5p − (p + 2) = 5p − 1(p + 2)= 5p − 1 × p − 1 × 2= 5p − p − 2= 4p − 2

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48 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 Expand:a 5(a + 2) b 6(m − 3) c 2(j + 9)d 4(x + y) e 9(p − q) f 11(m − 4)g 7(2k + 5) h 3(5m − 2)

2 Expand:a -3(k + 4) b -4(m − 1) c -7(r + 3)d -2(d + e) e -5(g − h) f -2(c + 3)g -9(5 + 2x) h -8(3f + 2)

3 Expand:a x(y + 3) b p(q − 7) c d(d + 4)d h(2h − 1) e a(a + 3c) f 2l (3m + 2n)g 5k(3k − 2) h 7j(3j + 5k)

4 When expanded, 3x(x − 4) is which of the following? Select A, B, C or D.A 3x + 12 B 3x2 − 7x C 3x2 − 12x D 3x + 7

5 Expand:a x(4x + y) b -2(4p − 3q)c k(k − 12) d -5(4 − m)e 4n(2n + 5m) f 2x(1 − 7x)g -f (2f + 4) h -(5 + 7k )

6 Explain the difference between 2a + 1 and 2(a + 1).

7 Expand and simplify:a 7(a + 2) + 12 b 6k + 5(k + 3)c 3(q − 2m) + 3m d 9 + 6(p − 3)e 18 − (3y + 14) f 21 − 6(t + 2)g 6(2p − 1) + 13 h 4y − 4(3y − 2)

8 Check that 8(2a − 3) = 16a − 24 by substituting any value for a and showing that the answers to both sides are equal.

9 Expand and simplify:a 5(a + 4) + 2(a + 1) b 2(x + 2) + 4(x − 1)c 7(4 + j) + 5(2j − 2) d 5(2t + 1) + 3(4t + 2)e 3(n + 5) − 2(n + 4) f 10(w + 2) − 6(w + 3)g 8(3c + 1) − 5(2c − 3) h 3(p − 2) − 3(p − 4)i m(m + 1) + 3(m + 1) j 4(d + e) + 3(d − e)k 3f(5f + 2) − 4(2f − 1) l 4e(1 − 2e) − e(3e + 4)

Exercise 2-08

Ex 11

Ex 12

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49ISBN: 9780170136952 CHAPTER 2 ALGEBRA

10 Jill says that, to find the perimeter of a rectangle, you double its length, double its breadth, and add the answers together. Thomas says that you add the length and breadth first, then double the answer. Who is correct? Can you write each of their answers algebraically?

2-09 Factorising algebraic termsThe factors of a number are those numbers that divide into it evenly. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.The highest common factor (HCF) of two numbers is the largest number that divides into both numbers evenly. For example, the HCF of 18 and 30 is 6.

Algebraic terms also have factors. Some of the factors of 10xy are 1, 2, 5, x, 2y and 10xy. Often there are too many to list them all.

Example 13

Find the highest common factor (HCF) of 48 and 20.

SolutionWrite the factors of each number.The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.The factors of 20 are: 1, 2, 4, 5, 10, 20.The common factors of 48 and 20 are: 1, 2 and 4.The highest common factor is 4.

Example 14

List four factors of 40a2b.

SolutionThere are many ways of multiplying terms to get 40a2b, such as:

8 × 5 × a × a × b4 × 10 × a2 × b40 × a × ab

So four factors of 40a2b are: 4, 10, a and ab. (Note: It is impractical to list them all.)

To find the highest common factor (HCF) of two or more algebraic terms:• find the HCF of the numbers• find the HCF of the variables• multiply them together to make the highest common algebraic factor.

!

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50 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 What are the factors of each of these?a 9 b 8 c 15d 25 e 6 f 30g 42 h 36 i 55j 24 k 16 l 27

2 Find the highest common factor of each of these pairs of numbers:a 9 and 6 b 6 and 14c 8 and 12 d 15 and 10e 18 and 24 f 16 and 12g 33 and 22 h 15 and 25i 9 and 21

3 Which of the following is not a factor of 36m2n? Select A, B, C or D.A 4m B 9n C 8mn D 12m2n

4 List four factors of each of these terms:a 5m b 7pqc 10x d 18de 4k2 f 21bcg 11x2y h 8ab

5 Find the highest common factor for each pair of terms below:a 3x and 12x b 64j and 24j c 6x and 4axd 21kj and 14k e 5pq and 10p f 16m and 12mng 48cd and 32bc h 20mn and 15np i 9x and 12x2

Exercise 2-09

Example 15

Find the highest common factor of 16x and 40xy.

SolutionFirst, find the HCF of the numbers:The factors of 16 are: 1, 2, 4, 8, 16.The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40.The HCF of 16 and 40 is 8.Then find the HCF of the variables:

The HCF of x and xy is x.∴ The HCF of 16x and 40xy is 8 × x = 8x.

Ex 13

Ex 14

Ex 15

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51ISBN: 9780170136952 CHAPTER 2 ALGEBRA

Mental skills 2

Simplifying multiplication by factorisingSometimes, one big multiplication may be made into many simpler little multiplications if we break one or both terms into two ‘easier’ factors. Then we use the property that numbers can be multiplied in any order.

1 Examine these examples:a 15 × 6 = 3 × 5 × 6 Look for pairs of numbers you can multiply easily.

= 3 × 30= 90

b 30 × 22 = 10 × 3 × 22= 10 × 66= 660

Note: For each question, there are many different possible ways of arriving at the correct result.

2 Now find these products by factorising first:a 30 × 24 b 18 × 27 c 25 × 33 d 16 × 8e 28 × 20 f 12 × 18 g 21 × 9 h 36 × 12i 16 × 35 j 22 × 28 k 9 × 15 l 27 × 25

Maths without calculators

Algebra is included as a muse in this Vatican artwork. Some of the other ‘muses’ included are Medicina (Medicine), Jurisprudentia

(Justice), Historia (History), Poesis (Poetry) and Musica (Music).

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52 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

2-10 Factorising algebraic expressionsFactorising is the opposite of expanding. To factorise an algebraic expression, such as 3x + 18, divide each term by a common factor and insert brackets.

Skillsheet2-02

Algebra usingdiagrams

Skillsheet2-03

Factorising using diagrams

Factorising an expression• Find the HCF of the terms and write it outside the brackets.• Divide each term by the HCF and write them inside the brackets.

ab + ac = a(b + c)ab – ac = a(b – c)

• To check that the factorised answer is correct, expand it.

!expanding

factorising

3(x + 6) = 3x + 18

Example 16

Factorise each of the following expressions:a 3x + 6 b 4 − 4x

Solutiona 3x + 6 = 3 × x + 3 × 2 The highest common factor of 3x and 6 is 3.

3x ÷ 3 = x 6 ÷ 3 = 2

= 3(x + 2) Check by expanding: 3(x + 2) = 3x + 6

b 4 − 4x = 4 × 1 − 4 × x

= 4(1 − x)

4 is the highest common factor of 4 and 4x.4 ÷ 4 = 1 4x ÷ 4 = x

Example 17

Factorise each of the following expressions:

a 2ab + 5b b x2 + 7x

Solutiona 2ab + 5b = b × 2a + b × 5 b is the highest common factor.

2ab ÷ b = 2a 5b ÷ b = 5

= b(2a + 5) Check by expanding: b(2a + 5) = 2ab + 5b

b x2 + 7x = x × x + 7 × x

= x(x + 7)

x is the highest common factor.x2 ÷ x = x 7x ÷ x = 7

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53ISBN: 9780170136952 CHAPTER 2 ALGEBRA

1 Copy and complete:a 2x + 4 = 2 (____) b 4p − 16 = 4 (____)c 12k − 15 = 3 (____) d 5j − 20k = 5 (____)e 3x2 + 3y2 = 3 (____) f 5x − 15 = ____ (x − 3)g 12x − 6 = ____ (2x − 1) h 2a + 8b = ____ (a + 4b)i 4pq + 10 = ____ (2pq + 5) j 5 + 15x = 5 (____)

2 Copy and complete:a xy + xz = x (____) b ab − ac = a (____)c pq − p = p (____) d cd + bc = c (____)e a2 + 2a = a (____) f 5hk − 8h = ____ (5k − 8)g 24mn + 11n = ____ (24m + 11) h xy + y2 = ____ (x + y)i 3k2 + 4jk = ____ (3k + 4j) j ef − 2f 2 = ____ (e − 2f )

3 Factorise each of the following expressions by finding the highest common factor:a 4x + 8 b 3m + 6c 5a − 10 d 15m − 10ae 8x − 4y f 12m − 16wg hk − h h 2xy − xmi 12p + 5pr j x2 + 5xk 4m − m2 l y2 − 11y

4 Which of the following is 16k2 + 18k factorised completely? Select A, B, C or D.A 2(8k2 + 9k)B k(16k + 18)C 2k(8k + 9)D 4k(4k + 9)

5 Copy and complete:a 2am + 2an = 2a(____) b 6xy − 3x = 3x(____)c 4ap + 2a = 2a(____) d 5mn − 5mp = 5m(____)e 3xy − 12x = 3x(____) f 2pq + 2pr = 2p(____)g 14x − 2xy = 2x(____) h 20kj + 5j = 5j(____)i 9xy − 12yz = 3y(____)

6 Factorise these:a 7ab + 14a b 2x − 6xy c 3m − 9mwd 10xy − 25xw e 8am − 12mp f 9xy + 6myg 3x + 9xy h ab + abc i xyz + xyaj 18m − 24mn k 2ac + 3abc l pqr + mpq

Exercise 2-10

Ex 16

Ex 17

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54 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

7 Factorise:a rs + 7r b 4s2 + 16c p2 − 2p d 6k3 − 2ke 3j + j2 f 7n − n2

g 7e + 21e2 h 6q − 12q2

i a4 + a2 j x2y + xy2

k 3m2n + 9m l 12a2b2c2 + 3abc

2-11 Factorising with negative termsWhen factorising an algebraic expression where the first term is negative, include the negative sign in the highest common factor.

1 Copy and complete:a -4k − 24 = -4 (_________) b -mn − mp = -m (_________)c -5r + 25 = -5 (_________) d -9k + 27 = -9 (_________)e -ap + aq = ____ (p − q) f -x2 − 7x = ____ (x + 7)g -12y − 18 = ____ (2y + 3) h -4f + f 2 = ____ (4 − f )

2 Factorise each of the following expressions:a -4x + 8 b -3m − 6c -5x + 10 d -15x + 10me -6x − 8m f -5x − 10g -a − ab h -3xyz + 6abci -16 + 4x j -ax − ayk -2m + 5mp l -24x − 16y

Exercise 2-11

Worksheet2-08

Expandominoes

Skillsheet2-02

Algebra usingdiagrams

Skillsheet2-03

Factorising using diagrams

Worksheet2-09

Factorising puzzle

Example 18

Factorise each of the following expressions:

a -4x − 8 b -pq + q2

Solutiona -4x − 8 = -4 × x + (-4) × 2

= -4(x + 2)

-4 is the ‘highest’ common factor.-4x ÷ (-4) = x -8 ÷ (-4) = 2Check by expanding: -4(x + 2) = -4x − 8

b -pq + q2 = -q × p + (-q) × (-q)

= -q(p − q)

-q is the ‘highest’ common factor.-pq ÷ (-q) = p q2 ÷ (-q) = -q

Worksheet2-10

Algebra review

Ex 18

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55ISBN: 9780170136952 CHAPTER 2 ALGEBRA

3 Factorise each of these:a -16xy − 24xk b -16x + 12xyc -8mw + 12m d -5x − 15xye -ab − abc f -xya + xywg -12k − 18kw h -4rt + 7rsti -20fg + 4g j -19ap − 38ak -abc + abcd l -abc − abcd

2-12 Index laws: Multiplication and divisionRemember:

Consider these examples:• p3 × p4 = (p × p × p) × (p × p × p × p)

= p7

• m5 × m = (m × m × m × m × m) × m Note: m = m1

= m6

A shorter way to do this is to add the powers:• p3 × p4 = p3+4 = p7 • m5 × m = m5+1 = m6

Consider these examples:

• p6 ÷ p2 = =

= p × p × p × p= p4

• k3 ÷ k = = Note: k = k1

= k × k= k2

A shorter way to do this is to subtract the powers:• p6 ÷ p2 = p6−2 = p4 • k3 ÷ k = k3−1 = k2

power orindex

x5base

When multiplying powers with the same base, add the powers.am � an � am�n !

p6

p2----- p p p p p p×××××

p p×-----------------------------------------------

k3

k----- k k k××

k---------------------

When dividing powers with the same base, subtract the powers.

am � an � � am�nam

an-------- !

Stage 5

02 NCM8_4 SB TXT.fm Page 55 Monday, September 29, 2008 1:05 AM

Page 23: Chapter 2

56 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 Simplify each of the following:a m2 × m4 b p5 × p2 c k × k9

d y4 × y6 e k2 × k f x7 × x2

g h5 × h5 h n2 × n6 i q4 × q2

j t11 × t10 k d4 × d10 × d2 l s4 × s8 × s12

m m2 × m3 × m4 n k9 × k2 × k o p7 × p2 × p5

2 Which of the following is a2b × a4b3 simplified? Select A, B, C or D.A a8b3

B a8b4

C a6b3

D a6b4

3 Simplify each of the following:a 3x2 × 4 b 5 × 2p2 c 11m4 × 3d 6k2 × 7k2 e 2y3 × 4y f u4 × 2u2

g 8e3 × 5e h a2b × ab3 i m4n2 × m3n5

j k2l × kl4 k f 5g4 × f 3g7 l 2p2q × 3p2qm 5c4d3 × 5c2d n 12uv3 × 4u3v o 10x8y5 × 8x7y9

Exercise 2-12

Example 19

Simplify:a p6 × p2 b 3m2n × 7m3n5

Solutiona p6 × p2 = p6+2

= p8

b 3m2n × 7m3n5 = 3 × m2 × n1 × 7 × m3 × n5

= 3 × 7 × m2+3 × n1+5

= 21m5n6

Example 20

Simplify:

a p12 ÷ p4 b c

Solution

a p12 ÷ p4 = p12−4

= p8

b = k6−1

= k5

c = × y6−2

= 6y4

k6

k----- 42y6

7y2-----------

k6

k----- 42y6

7y2----------- 42

7------

Ex 19

Stage 5

02 NCM8_4 SB TXT.fm Page 56 Friday, September 19, 2008 10:54 AM

Page 24: Chapter 2

57ISBN: 9780170136952 CHAPTER 2 ALGEBRA

4 Simplify each of the following:

a y6 ÷ y4 b m4 ÷ m2 c k2 ÷ k d

e n9 ÷ n f x7 ÷ x2 g h e5 ÷ e3

i p3 ÷ p3 j r8 ÷ r7 k m10 ÷ m4 ÷ m2 l k12 ÷ k8 ÷ k

m q2 × q3 ÷ q4 n o p

5 Simplify each of the following:

a 16k5 ÷ 8 b 40a4 ÷ 10 c d

e 20g4 ÷ 5g2 f 18p6 ÷ 3p3 g 48v4 ÷ 8v3 h

i e8f 4 ÷ e2f 3 j k l 66x4y2 ÷ 11xy

m n 42g5h6 ÷ 6gh2 o 32p5q4 ÷ 4p4q3 p

2-13 Index laws: Raising a power to a powerConsider these examples:

• (p6)2 = p6 × p6 • (d2)3 = d2 × d2 × d2

= p6+6 = d2+2+2

= p12 = d6

A shorter way to do this is to multiply the powers.• (p6)2 = p6×2 = p12

• (d2)3 = d2×3 = d6

Ex 20q5

q2-----

j11

j9------

c5 c2×c3

--------------- d4 d3×d2

----------------- a5

a5-----

25l7

5---------- 24r

12---------

54n5

9n------------

s9t7

s3t2--------- m12n16

m5n9------------------

80k7l5

20k3l---------------- 18d2e

21de2--------------

Worksheet2-11

Index laws

Worksheet2-12

Indices puzzle

When raising a term with a power to another power, multiply the powers.(am)n � am�n !

Example 21

Simplify:a (p2)5 b (d5)4 c (5m2)3 d (2n2p3)4

Solutiona (p2)5 = p2 × 5

= p10b (d5)4 = d5 × 4

= d20

Stage 5

02 NCM8_4 SB TXT.fm Page 57 Friday, September 19, 2008 10:54 AM

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58 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

1 Simplify each of the following:a (n9)2 b (k2)3 c (y4)5 d (x2)7 e (j8)3

f (b6)3 g (k7)2 h (v5)2 i (t10)2 j (a7)2

2 Which of the following is (a4b2)3 simplified? Select A, B, C or D.A a7b5 B a12b6 C a12b5 D a7b6

3 Simplify each of the following:a (2k)2 b (5m2)3 c (ab)3 d (p2q)5

e (3g4)4 f (m4n)5 g (2kj)3 h (4c2d)2

i (10r4s2)4 j (5x4y9)3 k (2m4p2)7 l (x2y3z5)6

Exercise 2-13

c (5m2)3 = 5m2 × 5m2 × 5m2

= 53 × m2×3

= 125m6

d (2n2p3)4 = 24 × n2×4 × p3×4

= 16n8p12

Ex 21

Just for the record

Alan Turing (1913–1954)Alan Turing was an English mathematician who was born in London and died at an early age from potassium cyanide poisoning during an electrolysis experiment.He was one of the pioneers of computer theory. During World War II he worked at the Government Code and Cipher School and, in 1945, was involved in designing and constructing a large automatic computer called ACE (Automatic Computing Engine). In 1949 he became deputy director of the laboratory where the largest memory-capacity computer in the world was built. The computer was called MADAM (Manchester Automatic Digital Machine). Turing put forward the idea that a computer could be built which could think like a human.

Find out what Turing achieved at the Code and Cipher School.

Stage 5

02 NCM8_4 SB TXT.fm Page 58 Friday, September 19, 2008 10:54 AM

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59ISBN: 9780170136952 CHAPTER 2 ALGEBRA

Power plus

1 Factorise each of the following:a 9pq + 5p − 2p2

b 44k − 33l − 11mc 2x − 7x2 + 3xyd x(y + 2) + 3 (y + 2)e a(b − 3) − 7 (b − 3)f 4(k + 1) − j (k + 1)

2 For each of these shapes, find an algebraic expression for:i the perimeter

ii the area.(Give your answer in simplest form.)

a b c

3 Simplify each of the following:

a

b

c

4 Find out how to expand:a (x + 3)(x + 2)b (2 + x)(x − 7)

5 Find out the meaning given to a number raised to the power zero.For example, x0 means ________ and 50 means ________.Explain why this is so and give more examples.

6 Find out the meaning given to negative powers.For example, x-1 means ________ and 10-2 means ________.Explain and give four examples.

7 Find out the meaning given to powers that are fractions.

For example, means ________ and means ________.Explain and give four examples.

x + 3

xp

x

y

16

4x--- 2

x---+

310x--------- 1

5x------–

7a--- 5

b---+

x12---

6413---

02 NCM8_4 SB TXT.fm Page 59 Friday, September 19, 2008 10:54 AM

Page 27: Chapter 2

60 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

Language of mathsalgebraic expression algebraic term base differenceevaluate expand factor factoriseformula grouping symbols highest common factor (HCF)index laws like terms pattern productpronumeral rule simplify substitutesubstitution variable

1 Why does the word ‘variable’ have that name? Look up its meaning in the dictionary.

2 What are unlike terms?

3 How do you expand an algebraic expression?

4 What is the name given to a number that divides evenly into another number or algebraic term?

5 What is the opposite of expanding an algebraic expression?

6 Look up some non-mathematical meanings of the word ‘factor’. Use ‘factor’ in a sentence to show one of these meanings.

Topic overview• What parts of algebra did you remember from Year 7? What new rules have you learned?• Are there parts of this chapter that you still do not understand? Talk to your teacher.• Give two examples of jobs where algebra is needed.• Copy and complete this chapter summary. Highlight what you often get wrong.

Worksheet2-10

Algebra review

ALGEBRA

Expanding

Add/subtractlike terms

Algebraicfractions

Multiplicationand division

Substitution

Factorising2x + 2y2(x + y)

Words toalgebra

Chapter 2 reviewWorksheet2-13

Algebra find-a-word

02 NCM8_4 SB TXT.fm Page 60 Friday, September 19, 2008 1:00 PM

Page 28: Chapter 2

61ISBN: 9780170136952 CHAPTER 2 ALGEBRA

Chapter revision1 Write each of these as an algebraic expression:

a three more than N b seven times A minus three c M divided by N

2 If x = 4 and y = -3, find the values of:a x + y b xy c 3x − 2y d y2 e 6x ÷ y f x2 + y2

3 Simplify each of the following:a 6k + 7k b 19d − 2d c 4p + 18p − 7pd 17m − 3m + 6m e 3pq + 4pq − 2pq f 6cd + 7cd − 3cdg 4a + 2b + 3a + 7b h 5r + 9s + 11r + 2s i 13f + 6g − 5f + g

4 Simplify each of the following:a 4 × 2x b 3b × 7c c 12m × 5 d 4p × 8q e 2 × 3a × 4a

f 25j ÷ 5 g h i p × p × p × p

5 Simplify each of the following:

a b c d e

6 Simplify each of the following:

a b c d e

7 Expand each of the following:a 4(a + 7) b 6(2k + 1) c -2(n + 4) d -8(4 + v)

8 Expand each of the following:a m(n + 4) b a(2a + 3b) c 7j(2j − 3)

9 Expand and simplify each of these:a 12 + 2(n + 4) b 12v − 3(v + 9) c 3(y + 1) + 5(y + 4)

10 Find the highest common factor of each of these pairs of terms:a 3a and 9 b 4p2 and 8p c 2t and 6

11 Factorise each of these expressions:a 3x + 9 b 5N − 25 c 6a + 9ab d 8N − 12NMe xy + xyz f ab + abc g y2 + 7y h 8kj + 12j2

12 Factorise each of these expressions:a -2 − 8q b -16r − 4 c -12 + 15t d -pq + pe -16x − 24xy f -k2 − 4k g -s2 + 7s h -m2n − mn2

13 Simplify:a x2 × x4 b m3 × m2 × m c 2p4 × p3 d 2pq2 × 4p2q2

14 Simplify:a n9 ÷ n3 b c 16l4 ÷ l2 d

15 Simplify:a (d4)2 b (pq)4 c (2m2)3 d (k4l3)6

Exercise 2-01

Exercise 2-02

Exercise 2-03

Exercise 2-04

32rs4

----------- 15p3p

---------

Exercise 2-05

k6--- 2k

6------+ 11m

8----------- 7m

8--------– 3d

4------ d

4---– f

4--- 3f

5-----+ 5s

6----- 3s

4-----–

Exercise 2-06

mn---- n

m----× 4m

5-------- 2

12m-----------× 12

x------ 2x

7------× k

4--- k

7---÷ 18

a------ 9

a---÷

Exercise 2-08

Exercise 2-08

Exercise 2-08

Exercise 2-09

Exercise 2-10

Exercise 2-11

Exercise 2-12

Exercise 2-12

v5

v2----- 48r4

8r3-----------

Exercise 2-13

Topic testChapter 2

02 NCM8_4 SB TXT.fm Page 61 Friday, September 19, 2008 1:00 PM